Local existence for the non­resistive MHD equations in nearly optimal Sobolev spaces

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Fefferman, Charles L, McCormick, David S, Robinson, James C and Rodrigo, Jose L (2017) Local existence for the non-resistive MHD equations in nearly optimal Sobolev spaces. Archive for Rational Mechanics and Analysis, 223 (2). pp. 677-691. ISSN 0003-9527

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Local existence for the non-resistive MHDequations in nearly optimal Sobolev spaces

Charles L. FeffermanDepartment of Mathematics, Princeton University,

Fine Hall, Washington Road, Princeton, NJ 08544. USA.

David S. McCormickSchool of Mathematical and Physical Sciences,

Pevensey 2, University of Sussex, Brighton BN1 9QH. UK.

James C. Robinson & Jose L. RodrigoMathematics Institute, University of Warwick,

Coventry CV4 7AL. UK.

August 9, 2016

Abstract

This paper establishes the local-in-time existence and uniqueness ofsolutions to the viscous, non-resistive magnetohydrodynamics (MHD)equations in Rd, d = 2, 3, with initial data B0 ∈ Hs(Rd) and u0 ∈Hs−1+ε(Rd) for s > d/2 and any 0 < ε < 1. The proof relies onmaximal regularity estimates for the Stokes equation. The obstructionto taking ε = 0 is explained by the failure of solutions of the heatequation with initial data u0 ∈ Hs−1 to satisfy u ∈ L1(0, T ;Hs+1); weprovide an explicit example of this phenomenon.

1 Introduction

In this paper we consider the equations of MHD with zero magnetic resistiv-ity,

ut −∆u+ (u · ∇)u+∇p = (B · ∇)B, ∇ · u = 0, (1.1a)

Bt + (u · ∇)B = (B · ∇)u, ∇ ·B = 0, (1.1b)

1

along with specified initial data u(0) = u0 and B(0) = B0. Jiu & Niu(2006) established the local existence of solutions in 2D for initial data inHs for integer s ≥ 3, and more recently Ren et al. (2014) and Lin et al.(2015) have established the existence of global-in-time solutions for initialdata sufficiently close to certain equilibrium solutions (again in 2D).

In a previous paper (Fefferman et al., 2014) we proved a local existenceresult for these equations taking arbitrary initial data in u0, B0 ∈ Hs(Rd)with s > d/2 for d = 2, 3. However, given the presence of the diffusive termin the equation for u, it is natural to expect that such a local existence resultshould be possible with less regularity for u0 than for B0.

To this end, it was shown in Chemin et al. (2016) that one can prove local

existence when B0 ∈ Bd/22,1 (Rd) and u0 ∈ Bd/2−1

2,1 (Rd). The underlying observa-

tion that allowed for such a result is that when u0 ∈ Bd/2−12,1 (Rd) the solution u

of the heat equation with initial data u0 is an element of L1(0, T ;Bd/2+12,1 (Rd)):

in these Besov spaces the solution regularises sufficiently that an additionaltwo derivatives become integrable in time.

In Sobolev spaces this does not occur: in Lemma 2.1 we show that foru0 ∈ Hs we have

u ∈ L∞(0, T ;Hs) ∩ L2(0, T ;Hs+1) ∩ Lq(0, T ;Hs+2)

for any 0 < q < 1, and this is in some sense the best possible. The failureof the estimate u ∈ L1(0, T ;Hs+2) is ‘well known’, but it is not easy to findany explicit example in the literature, so we provide one here in Lemma 2.2.

In this paper we therefore take B0 ∈ Hs(Rd), with s > d/2, and u0 slightlymore regular than Hs−1(Rd), namely u0 ∈ Hs−1+ε(Rd), with 0 < ε < 1. Bymaking use of maximal regularity results for the heat equation (which werecall in the next section) we are then able to prove the local existence of asolution that remains bounded in these spaces (see Theorem 3.1 for a precisestatement).

Throughout the paper we use the notation Λs to denote the fractionalderivative of order s, given in terms of the Fourier transform by

Λsf = |ξ|sf .

We write‖u‖2

Hs = ‖Λsu‖2 + ‖u‖2, s > 0,

which is equivalent to the standard Hs norm when s is a positive integer.

2

2 Estimates for solutions of the heat equation

in Sobolev spaces

2.1 Energy estimates

First we prove some standard estimates for solutions of the heat equation,including the Lq(0, T ;Hs+2) estimate for 0 < q < 1. We give the proofs,since we will need to keep careful track of the dependence of the estimateson T ; for simplicity we restrict to T ≤ 1, which will of course be sufficientfor local existence arguments.

We use relatively elementary energy methods, but the results can bederived by other means; for example, the fact that u ∈ Lq(0, T ;Hs+2) followsfrom the smoothing estimate ‖∂βe∆tf‖L2 ≤ Ct−|β|/2‖f‖L2 , as kindly pointedout by the referee.

Lemma 2.1. If u0 ∈ Hs(Rd) and u denotes the solution of the heat equation

∂tu = ∆u with u(0) = u0

then u ∈ L∞(0, T ;Hs)∩L2(0, T ;Hs+1) and√tu ∈ L2(0, T ;Hs+2(Rd)). When

T ≤ 1

sup0≤t≤T

‖u(t)‖2Hs ,

∫ T

0

‖u(t)‖2Hs+1 ,

∫ T

0

t‖u(t)‖2Hs+2 ≤ ‖u0‖2

Hs .

Consequently u ∈ Lq(0, T ;Hs+2(Rd)) for any 0 < q < 1, and∫ T

0

‖u(t)‖qHs+2 ds ≤ CqT1−q‖u0‖qHs (2.1)

provided that T ≤ 1.

Proof. We start with the L2 estimate obtained by taking the inner productwith u in L2 to give

1

2

d

dt‖u‖2 + ‖∇u‖2 = 0

and then integrating in time,

1

2‖u(t)‖2 +

∫ t

0

‖∇u(τ)‖2 dτ =1

2‖u0‖2.

To bound the higher derivatives we act on the equations with Λs and thentake the L2 inner product with Λsu to obtain

1

2

d

dt‖Λsu‖2 + ‖Λs+1u‖2 = 0,

3

followed by an integration in time,

1

2‖Λsu(t)‖2 +

∫ t

0

‖Λs+1u(τ)‖2 dτ =1

2‖Λsu0‖2. (2.2)

Combining these two estimates shows that u ∈ L∞(0, T ;Hs)∩L2(0, T ;Hs+1),with

sup0≤t≤T

‖u(t)‖2Hs ≤ ‖u0‖2

Hs

and ∫ T

0

‖u‖2Hs+1 ≤

1

2‖Λsu0‖2 + T‖u0‖2;

in particular if T ≤ 1 then∫ T

0

‖u‖2Hs+1 ≤ ‖u0‖2

Hs .

To obtain the bound on√tu in Hs+2 we act on the equations with Λs+1

and take the L2 inner product with tΛs+1u. Then

1

2

d

dt

(t‖Λs+1u‖2

)− 1

2‖Λs+1u‖2 + t‖Λs+2u‖2 = 0.

Integrating from 0 to T yields

T

2‖Λs+1u(T )‖2 +

∫ T

0

t‖Λs+2u(t)‖2 dt ≤ 1

2

∫ T

0

‖Λs+1u(t)‖2 dt ≤ 1

4‖Λsu0‖2,

(2.3)using the bound from (2.2). We also have∫ T

0

t‖u(t)‖2 dt ≤ T 2

2‖u0‖2,

from which it follows that√tu ∈ L2(0, T ;Hs+2). For T ≤ 1 we can combine

this with (2.3) to obtain the estimate∫ T

0

t‖u‖2Hs+2 dt ≤ 1

2‖u0‖2

Hs .

For any q < 1 we have, using Holder’s inequality with exponents 2/q and2/(2− q),∫ T

0

‖u(t)‖qHs+2 =

∫ T

0

t−q/2tq/2‖u(t)‖qHs+2

≤(∫ T

0

t−q/(2−q))1−(q/2)(∫ T

0

t‖u(t)‖2Hs+2

)q/2≤ CqT

1−q‖u0‖qHs ,

since 0 < q < 1 ensures that q/(2−q) < 1 and the first term is integrable.

4

2.2 An example of u0 ∈ L2 with u /∈ L1(0, T ;H2)

We now provide an explicit example of an initial condition u0 ∈ L2(Rd) suchthat u /∈ L1(0, T ;H2(Rd)).

Lemma 2.2. There exists u0 ∈ L2(Rd) such that the solution u of the heatequation with initial data u0 is not an element of L1(0, T ;H2(Rd)).

Proof. We let u0 be the function with Fourier transform

u0(ξ) =1

|ξ|d/2 log(2 + |ξ|). (2.4)

The solution u(t) of the heat equation with initial data u0 has Fourier trans-form

u(t, ξ) = u0(ξ)e−|ξ|2t

and

‖u(t)‖2H2 =

∫Rd

|ξ|4|u0(ξ)|2e−2|ξ|2t dξ.

We therefore have

I :=

∫ T

0

‖u(t)‖H2 dt =

∫ T

0

(∫Rd

|ξ|4|u0(ξ)|2e−2|ξ|2t dξ

)1/2

dt.

In order to bound this from below we split the range of time integration,choosing j0 such that j−2

0 ≤ T , and write

I ≥N∑j=j0

∫ j−2

(j+1)−2

(∫Rd

|ξ|4|u0(ξ)|2e−2|ξ|2t dξ

)1/2

dt

≥N∑j=j0

∫ j−2

(j+1)−2

(∫|ξ|≤j|ξ|4|u0(ξ)|2e−2|ξ|2t dξ

)1/2

dt

≥ e−1

N∑j=j0

∫ j−2

(j+1)−2

(∫|ξ|≤j|ξ|4|u0(ξ)|2 dξ

)1/2

dt

≥ e−1

N∑j=j0

1

j2(j + 1)

(∫|ξ|≤j|ξ|4|u0(ξ)|2 dξ

)1/2

.

5

By our choice of u0(ξ) in (2.4) we have∫|ξ|≤j|ξ|4|u0(ξ)|2 dξ =

∫|ξ|≤j|ξ|4 1

|ξ|d[log(2 + |ξ|)]2dξ

≥ 1

[log(2 + j)]2

∫|ξ|≤j|ξ|4−d dξ

≥ c

[log(2 + j)]2j4.

It follows that

I ≥ ce−1

N∑j=j0

1

(j + 1) log(2 + j),

as since the sum is unbounded as N →∞ it follows that u /∈ L1(0, T ; H2) asclaimed.

2.3 Maximal regularity-type results

Usually, ‘maximal regularity’ results for the heat equation yield

∂tu, ∆u ∈ Lp(0, T ;Lq)

when u solves∂tu−∆u = f, u(0) = 0

with f ∈ Lp(0, T ;Lq). The results follows from maximal regularity whenp = q, obtained as inequality (3.1) in Chapter IV, Section 3, pages 289–290of Ladyzhenskaja et al. (1968) using the Mihlin Multiplier Theorem (Mihlin,1957) and then an interpolation theorem due to Benedek et al. (1962); theprocedure is clearly explained in Krylov (2001), for example. However, inthis paper we will only require the following L2-based Sobolev-space result,for which the basic L2(0, T ;L2) maximal regularity estimate can be obtainedrelatively easily.

Proposition 2.3. There exists a constant Cr such that if f ∈ Lr(0, T ;Hs),s ≥ 0, and u satisfies

∂tu−∆u = f, u(0) = 0,

then u ∈ Lr(0, T ; Hs+2) with

‖u‖Lr(0,T ;Hs+2) ≤ Cr‖f‖Lr(0,T ;Hs). (2.5)

The constant Cr can be chosen uniformly for all 0 ≤ T ≤ 1.

6

Proof. First we treat the case s = 0, i.e. we bound u ∈ Lr(0, T ;H2) in termsof f ∈ Lr(0, T ;L2). The passage from estimates for the case r = 2 to thecase 1 < r < ∞ is covered by Krylov (2001). We therefore only prove theestimates in the case r = 2.

The L2 norm of u is bounded simply by taking the L2 inner product withu, using the Cauchy-Schwarz inequality, and integrating in time,

1

2

d

dt‖u‖2 + ‖∇u‖2 = 〈f, u〉 ≤ ‖f‖‖u‖ ≤ 1

2‖u‖2 +

1

2‖f‖2

and sod

dt[e−t‖u‖2] ≤ e−t‖f(t)‖2,

which implies (since u(0) = 0) that

‖u(t)‖2 ≤∫ t

0

‖f(s)‖2et−s ds,

whence ∫ T

0

‖u(t)‖2 ≤∫ T

0

∫ t

0

‖f(s)‖2et−s ds dt

=

∫ T

0

∫ T

s

‖f(s)‖2et−s dt ds ≤ eT∫ T

0

‖f(s)‖2 ds,

i.e.‖u‖L2(0,T ;L2) ≤ eT‖f‖L2(0,T ;L2). (2.6)

For ‖u‖H2 we can argue directly from the Fourier transform of the solutionu, since

Λ2u(ξ, t) =

∫ t

0

|ξ|2e−|ξ|2(t−s)f(s, ξ) ds.

Define

G(ξ, t) =

{|ξ|2e−|ξ|

2t t ≥ 0

0 t < 0

and

F (ξ, t) =

{f(ξ; t) 0 ≤ t ≤ T

0 otherwise.

Then

Λ2u(ξ; t) =

∫RG(ξ, t− s)F (ξ, s) ds

7

and we can use Young’s inequality for convolutions to give

‖Λ2u(ξ, ·)‖L2(0,T ) ≤ ‖Λ2u(ξ, ·)‖L2(R) ≤ ‖F (ξ, ·)‖L2(R) = ‖f(ξ, ·)‖L2(0,T ),

since ‖G(ξ, ·)‖L1 = 1. Therefore

‖u‖L2(0,T ;H2) = ‖Λ2u‖L2((0,T )×Rd) ≤ ‖f‖L2((0,T )×Rd) = ‖f‖L2(0,T ;L2).

Combined with (2.6) this yields ‖u‖L2(0,T ;H2) ≤ C2‖f‖L2(0,T ;L2), and hence(via the results of Benedek et al. (1962)) we obtain

‖u‖Lr(0,T ;H2) ≤ Cr‖f‖Lr(0,T ;L2).

Letting Cr be the constant for the choice T = 1, it can easily be seen thatthis constant is also valid for T ≤ 1 by extending f to be zero on the interval(T, 1].

We now apply this estimate to u = v and u = Λsv: we obtain

‖v‖Lr(0,T ;H2) ≤ Cr‖f‖Lr(0,T ;L2)

and

‖v‖Lr(0,T ;Hs+2) = ‖Λsv‖Lr(0,T ;H2) ≤ Cr‖Λsf‖Lr(0,T ;L2) = Cr‖f‖Lr(0,T ;Hs),

from which (2.5) follows.

We will apply this result in combination with the regularity results forthe heat equation from Lemma 2.1 in the following form for solutions of theStokes equation, allowing for non-zero initial data. Note that in order toobtain an L1-in-time estimate on ‖u‖Hs+1 we require Lr integrability of fwith r > 1, and the initial data to be in Hs−1+ε. Considering the equationsin Besov spaces as in Chemin et al. (2016) allows r = 1 and ε = 0 (ands = d/2 rather than s > d/2 in the final results).

Corollary 2.4. If f ∈ Lr(0, T ;Hs−1), 1 < r <∞, s > 1, and

∂tu−∆u+∇p = f, ∇ · u = 0, u(0) = u0 ∈ Hs−1+ε,

where u0 is divergence free, then for T ≤ 1∫ T

0

‖u‖Hs+1 ≤ CεTε/2‖u0‖Hs−1+ε + CrT

1− 1r ‖f‖Lr(0,T ;Hs−1). (2.7)

8

Proof. First we consider the solution v of

∂tv −∆v +∇π = 0, ∇ · v = 0, v(0) = u0.

Since u0 is divergence free, if we apply the Leray projector P (orthogonalprojection onto elements of L2 whose weak divergence is zero) we obtain

∂tv −∆v = 0, v(0) = u0,

since P commutes with derivatives on the whole space. It follows that v is infact the solution of the heat equation with initial data u0. We can thereforeuse Lemma 2.1 to ensure that

v ∈ L∞(0, T ;Hs−1+ε) ∩ L2(0, T ;Hs+ε) ∩ Lq(0, T ;Hs+ε+1)

for any q < 1, with all these norms depending only on the norm of the initialdata in Hs−1+ε. It follows by interpolation that v ∈ L1(0, T ;Hs+1) with∫ T

0

‖v‖Hs+1 ≤∫ T

0

‖v‖1−εHs+1+ε‖v‖εHs+ε

≤(∫ T

0

‖v‖2(1−ε)/(2−ε)Hs+1+ε

)(2−ε)/2(∫ T

0

‖v‖2Hs+ε

)ε/2,

using Holder’s inequality with exponents (2/(2− ε), 2/ε). Noting that

2(1− ε)2− ε

= 1− ε

2− ε< 1

we can use Lemma 2.1 to obtain∫ T

0

‖v‖Hs+1 ≤ CεTε/2‖u0‖Hs−1+ε . (2.8)

The difference w = u− v satisfies

∂tw −∆w +∇θ = f, ∇ · w = 0, w(0) = 0.

Again we can apply the Leray projector P, which is bounded from Hs−1 intoHs−1, to obtain

∂tw −∆w = Pf, w(0) = 0.

By the maximal regularity results for the heat equation from Proposition 2.3,we know that for any r > 1 we have∫ T

0

‖w‖Hs+1 ≤ T 1/r′‖w‖Lr(0,T ;Hs+1)

≤ CrT1/r′‖Pf‖Lr(0,T ;Hs−1)

≤ CrT1/r′‖f‖Lr(0,T ;Hs−1), (2.9)

where (r, r′) are conjugate. The inequality in (2.7) now follows by combining(2.8) and (2.9).

9

3 Proof of local existence

The main part of the proof consists of a priori estimates, which we proveformally. To make the proof rigorous requires some approximation proce-dure such as that employed in Fefferman et al. (2014), to which we refer forthe details. Where the limiting process involved would turns equalities intoinequalities, we write inequalities even in these formal estimates.

Theorem 3.1. Let d = 2, 3. Take s > d/2 and 0 < ε < 1. Suppose thatB0 ∈ Hs(Rd) and u0 ∈ Hs−1+ε(Rd). Then there exists T∗ > 0 such that thenon-resistive MHD system (1.1) has a solution (u,B) with

u ∈ L∞(0, T∗;Hs−1+ε(Rd)) ∩ L2(0, T∗;H

s+ε(Rd)) ∩ L1(0, T∗;Hs+1(Rd))

andB ∈ L∞(0, T∗;H

s(Rd)).

Note that the case ε = 1 was covered in a previous paper (Fefferman et al.2014), and is specifically excluded here.

Proof. Throughout the proof various constants will depend on s, but we donot track this dependency.

We first obtain a basic energy estimate in L2. We take the L2 innerproduct of the u equation with u and of the B equation with B to obtain

1

2

d

dt‖u‖2 + ‖∇u‖2 = 〈(B · ∇)B, u〉 and

1

2

d

dt‖B‖2 = 〈(B · ∇)u,B〉.

Since 〈(B · ∇)u,B〉 = −〈(B · ∇)u,B〉 we can add the two equations to yield

1

2

d

dt

(‖u‖2 + ‖B‖2

)+ ‖∇u‖2 ≤ 0

and so

‖u(t)‖2 + ‖B(t)‖2 + 2

∫ t

0

‖∇u(s)‖2 ds ≤ ‖u0‖2 + ‖B0‖2 =: M0. (3.1)

It is also helpful to have two other estimates for later use; observing that

|〈(B · ∇)u,B〉| ≤ c‖B‖‖∇u‖‖B‖L∞

10

and using the embedding Hs ⊂ L∞ (valid since s > d/2) and Young’s in-equality we obtain

d

dt‖u‖2 + ‖∇u‖2 ≤ c‖B‖2‖B‖2

Hs ≤ c‖B‖4Hs ; (3.2)

and similarly, since |〈(B · ∇)u,B〉| ≤ ‖B‖2‖∇u‖L∞ ,

1

2

d

dt‖B‖2 ≤ c‖∇u‖Hs‖B‖2. (3.3)

In order to estimate the norm of B in Hs we act on the B equation withΛs and take the inner product with ΛsB in L2. This yields

1

2

d

dt‖ΛsB‖2 ≤ |〈Λs[(B · ∇)u],ΛsB〉|+ |〈Λs[(u · ∇)B],ΛsB〉|

≤ c‖∇u‖Hs‖B‖2Hs (3.4)

using the fact that Hs is an algebra since s > d/2, along with the estimateproved in Fefferman et al. (2014)

|〈Λs[(u · ∇)B],ΛsB〉| ≤ c‖∇u‖Hs‖B‖2Hs ,

valid when s > d/2. Combined with (3.3) this yields

1

2

d

dt‖B‖2

Hs ≤ c‖∇u‖Hs‖B‖2Hs . (3.5)

The estimates for the u equation are more delicate. First we obtainestimates on u in the space L1(0, T ;Hs+1), using the maximal regularityestimates from Proposition 2.3 and Corollary 2.4. We consider the equationfor u as the forced Stokes equation

∂tu−∆u+∇p = f := −(u · ∇)u+ (B · ∇)B, ∇ · u = 0, u(0) = u0,

and so estimate (2.7) from Corollary 2.4 yields∫ T

0

‖u‖Hs+1 ≤ CεTε/2‖u0‖Hs−1+ε + CrT

1− 1r ‖f‖Lr(0,T ;Hs−1). (3.6)

Since

‖f‖Hs−1 = ‖∇ · (B ⊗B)−∇ · (u⊗ u)‖Hs−1

≤ ‖B ⊗B‖Hs + ‖u⊗ u‖Hs

≤ c‖B‖2Hs + c‖u‖2

Hs

≤ c‖B‖2Hs + c‖u‖2ε/(s+ε)‖u‖2s/(s+ε)

Hs+ε

≤ ‖B‖2Hs + cM

ε/(s+ε)0 ‖u‖2s/(s+ε)

Hs+ε ,

11

using (3.1), if we choose r = (s+ ε)/s > 1 then from (3.6) we have∫ T

0

‖u‖Hs+1 ≤ CεTε/2‖u0‖Hs−1+ε

+ CεTε/(s+ε)

(∫ T

0

‖B‖2(s+ε)/sHs + cM

ε/s0 ‖u‖2

Hs+ε dτ

)s/(s+ε).

(3.7)

We now estimate the norm of u in Hs−1+ε and Hs+ε. If we act withΛs−1+ε on the u equation and take the inner product with Λs−1+εu, then

1

2

d

dt‖Λs−1+εu‖2 + ‖Λs+εu‖2

≤ −〈Λs−1+ε[(u · ∇)u],Λs−1+εu〉+ 〈Λs−1+ε[(B · ∇)B],Λs−1+εu〉. (3.8)

For the first term on the right-hand side, we write

|〈Λs−1+ε[(u · ∇)u],Λs−1+εu〉| = |〈Λs−1[(u · ∇)u],Λs−1+2εu〉|= |〈Λs−1[∇ · (u⊗ u)],Λs−1+2εu〉|≤ c‖u‖2

Hs‖u‖Hs−1+2ε

≤ c(‖u‖εHs−1+ε‖u‖1−εHs+ε)

2‖u‖1−εHs−1+ε‖u‖εHs+ε

≤ c‖u‖1+εHs−1+ε‖u‖2−ε

Hs+ε

≤ c‖u‖2(1+ε)/ε

Hs−1+ε +1

4‖u‖2

Hs+ε ,

where we have used Sobolev interpolation, Young’s inequality, and the factthat Hs is an algebra (as s > d/2). The second term is handled similarly:

|〈Λs−1+ε[(B · ∇)B],Λs−1+εu〉| = |〈Λs−1[(B · ∇)B],Λs−1+2εu〉|= |〈Λs−1[∇ · (B ⊗B)],Λs−1+2εu〉|≤ c‖B‖2

Hs‖u‖Hs−1+2ε

≤ c‖B‖2Hs‖u‖1−ε

Hs−1+ε‖u‖εHs+ε

≤ c‖B‖2(1+ε)Hs + c‖u‖2(1+ε)/ε

Hs−1+ε +1

4‖u‖2

Hs+ε ,

using the three-term Young’s inequality with exponents (1 + ε, 2(1+ε)ε(1−ε) ,

2ε).

Combining these yields1

1

2

d

dt‖Λs−1+εu‖2 + ‖Λs+εu‖2 ≤ c‖B‖2(1+ε)

Hs + c‖u‖2(1+ε)/ε

Hs−1+ε +1

2‖u‖2

Hs+ε .

1Note that the exponent 2(1 + ε)/ε on the Hs−1+ε norm of u is far from optimal, andcan be reduced to some γ for 2 < γ ≤ 4 by using Lemma 1.1(i) from Chemin (1992).However, the proof here is significantly simpler, and still yields a short-time existenceresult (albeit with a possibly shorter existence time).

12

If we add (3.2) and an additional term +‖u‖2 to both sides then we obtain

1

2

d

dt‖u‖2

Hs−1+ε + ‖u‖2Hs+ε

≤ c‖u‖2(1+ε)/ε

Hs−1+ε + c‖B‖2(1+ε)Hs + c‖B‖4

Hs +1

2‖u‖2

Hs+ε + ‖u‖2,

and so

d

dt‖u‖2

Hs−1+ε+‖u‖2Hs+ε ≤ c1‖u‖2(1+ε)/ε

Hs−1+ε +c2‖B‖2(1+ε)Hs +c3‖B‖4

Hs+2‖u‖2. (3.9)

We now have three ingredients: the differential inequality (3.9) for u; theB equation (3.5)

1

2

d

dt‖B‖2

Hs ≤ c4‖∇u‖Hs‖B‖2Hs ,

which implies that

‖B(t)‖2Hs ≤ ‖B0‖2

Hs exp

(2c4

∫ t

0

‖∇u‖Hs dτ

); (3.10)

and the maximal regularity estimate∫ T

0

‖u‖Hs+1 ≤ CεTε/2‖u0‖Hs−1+ε

+ CεTε/(s+ε)

(∫ T

0

‖B‖2(s+ε)/sHs + c5M

ε/s0 ‖u‖2

Hs+ε dτ

)s/(s+ε).

(3.11)

We will now choose T ∗ such that ‖B(t)‖Hs ≤ 2‖B0‖Hs for all t ∈ [0, T∗].Set

M1 := ‖u0‖Hs−1+ε ,

and M2 := 22(1+ε)c2‖B0‖2(1+ε)Hs + 24c3‖B0‖4

Hs + 2M0,

and choose T ∗ sufficiently small that

0 <

(1− c1T (M2

1 + TM2)1/ε

ε

)−ε< 2 for all 0 < T < T∗ (3.12)

and

CεTε/2M1 + CεT

ε/(s+ε)(

22(s+ε)/sT‖B0‖2(s+ε)/sHs

+ c5Mε/s0

[c1T [2(M2

1 + TM2)](1+ε)/ε + TM2

] )s/(s+ε)<

log 4

2c4

(3.13)

13

for all 0 < T < T∗.To show that ‖B(t)‖Hs ≤ 2‖B0‖Hs for t ∈ [0, T∗], we make the assumption

that t 7→ ‖B(t)‖Hs is a continuous function that takes the value ‖B0‖Hs attime t = 0. While we have not shown this as part of our formal calculations,it would be true for any member of the family of smooth approximationsconsidered in Fefferman et al. (2014), and the estimates we now obtain wouldhold uniformly (for this family) for all t ∈ [0, T ∗] for the time T ∗ defined by(3.12) and (3.13).

Set

T = sup {T0 ∈ [0, T ∗] : ‖B(t)‖Hs ≤ 2‖B0‖Hs for all t ∈ [0, T0]}

and suppose that T < T ∗. Then from (3.1), (3.2), and (3.9) we obtain

d

dt‖u‖2

Hs−1+ε + ‖u‖2Hs+ε

≤ c1‖u‖2(1+ε)/ε

Hs−1+ε + 22(1+ε)c2‖B0‖2(1+ε)Hs + 24c3‖B0‖4

Hs + 2M0

≤ c1‖u‖2(1+ε)/ε

Hs−1+ε +M2 (3.14)

for all t ∈ [0, T ]. Using standard ODE comparison techniques (see, for ex-ample, Theorem 6 in Blomker et al. (2015), where we take p = (1 + ε)/ε) weobtain the bound

‖u(t)‖2Hs−1+ε ≤ (M2

1 + TM2)

(1− c1T (M2

1 + TM2)1/ε

ε

)−ε≤ 2(M2

1 + TM2) (3.15)

for all t ∈ [0, T ], by (3.12).Now, substituting (3.15) into (3.14) and integrating between times 0 and

T yields ∫ T

0

‖u(t)‖2Hs+ε dt ≤ c1T [2(M2

1 + TM2)](1+ε)/ε + TM2. (3.16)

Substituting (3.16) and ‖B(t)‖Hs ≤ 2‖B0‖Hs into (3.11), we obtain∫ T

0

‖u(t)‖Hs+1 dτ ≤ CεTε/2M1 + CεT

ε/(s+ε)(

22(s+ε)/sT‖B0‖2(s+ε)/sHs

+ c5Mε/s0

[c1T [2(M2

1 + TM2)](1+ε)/ε + TM2

] )s/(s+ε)<

log 4

2c4

, (3.17)

14

using (3.13).Substituting this into (3.10) ensures that ‖B(t)‖Hs < 2‖B0‖Hs for all

t ∈ [0, T ], contradicting the maximality of T . It follows that T = T ∗ andhence

‖B(t)‖Hs ≤ 2‖B0‖Hs for all t ∈ [0, T ∗].

The result now follows from (3.15), (3.16), and (3.17).

Conclusion

In the scale of Sobolev spaces we suspect that the result that we have provedhere is optimal. Bourgain & Li (2015) showed that the Euler equations onRd are ill posed in H1+d/2 for n = 2, 3, and we have shown via an explicitexample that for the heat equation we cannot gain the time integrability oftwo additional derivatives that is required in our local existence argument. Itwould be interesting to find a simpler model problem in which it is possibleto demonstrate the failure of local existence for B0 ∈ Hs and u0 ∈ Hs−1.

Conflict of Interest

The authors declare that they have no conflict of interest.

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